Mathematical models of the medulla of the mammalian kidney consist of complicated systems of nonlinear differential equations along with boundary conditions which are determined by the geometry of the flow tubes. The unknowns in these equations are, for each flow tube, the concentrations of several solutes, the volume flow rate, and the hydrostatic pressure. The primary objective of this research project is to extend these mathematical models so that the whole kidney, with its complete multinephron system, is represented by boundary value problems. The objective then will be to develop the mathematical theory for these renal flow problems. This theory will include a balance among existence criteria, uniqueness-nonuniqueness studies, bounds for solutions, stability analysis, and numerical analysis. The analysis will start by filling gaps in recent results given for simplifed kidney models. In particular, a more complete analysis will be developed for systems of flow tubes consisting of four, five, and six tubes where variable pressures and active transports are included. Then extensions of these models will be made to include several solutes and to include many tubes having a geometry that represents the whole kidney. Methodology will include different fixed point arguments, reducing the problems to corresponding singular perturbation problems and studying them, and studying the problems with different representations for the unknown concentrations, flow rates, and pressures. An effort will be made to use standard numerical techniques to develop the numerical analysis for the singular perturbation problems and then, using these results, to make numerical studies of the original problems. These numerical studies will include the application of some newer techniques that have recently appeared in the literature.